Regularized extended-hydrodynamic equations for a rarefied granular gas and the plane shock waves

2020 ◽  
Vol 5 (4) ◽  
Author(s):  
M. H. Lakshminarayana Reddy ◽  
Meheboob Alam
Keyword(s):  
Shock Waves ◽  
Plane Shock ◽  
Hydrodynamic Equations ◽  
Granular Gas

Plane shock waves and Haff’s law in a granular gas

2015 ◽  
Vol 779 ◽  
Author(s):  
M. H. Lakshminarayana Reddy ◽  
Meheboob Alam
Keyword(s):  
Mach Number ◽  
Shock Waves ◽  
Energy Equation ◽  
Navier Stokes ◽  
Maximum Temperature ◽  
Plane Shock ◽  
Density Peak ◽  
Granular Gas ◽  
Planar Shock ◽  
Weak Shocks

The Riemann problem of planar shock waves is analysed for a dilute granular gas by solving Euler- and Navier–Stokes-order equations numerically. The density and temperature profiles are found to be asymmetric, with the maxima of both density and temperature occurring within the shock layer. The density peak increases with increasing Mach number and inelasticity, and is found to propagate at a steady speed at late times. The granular temperature at the upstream end of the shock decays according to Haff’s law (${\it\theta}(t)\sim t^{-2}$), but the downstream temperature decays faster than its upstream counterpart. Haff’s law seems to hold inside the shock up to a certain time for weak shocks, but deviations occur for strong shocks. The time at which the maximum temperature deviates from Haff’s law follows a power-law scaling with the upstream Mach number and the restitution coefficient. The origin of the continual build-up of density with time is discussed, and it is shown that the granular energy equation must be ‘regularized’ to arrest the maximum density.

Shock waves in an ultra-relativistic fluid

1964 ◽  
Vol 277 (1368) ◽  
pp. 24-31
Keyword(s):  
Equation Of State ◽  
Shock Waves ◽  
Mass Density ◽  
Special Theory ◽  
Plane Shock ◽  
Theory Of Relativity ◽  
Special Theory Of Relativity ◽  
Conservation Equations ◽  
Relativistic Fluid ◽  
Density Relation

The special theory of relativity is used to analyze the motion of plane shock waves in a medium whose equation of state is u = 3 p , u being the mass density and p the pressure. The appropriate conservation equations together with this pressure-density relation provide a determinate set of equations for obtaining the downstream, in terms of the upstream, variables. The properties of normal and oblique shock waves in this gas are studied in the Lorentz frames in which the shocks are at rest.

ATTENUATION OF SPURIOUS OSCILLATIONS IN THE NUMERICAL CAPTURE OF SHOCK WAVES

2006 ◽  
Vol 17 (10) ◽  
pp. 1403-1413
Author(s):  
D. PORTES ◽  
H. RODRIGUES ◽  
S. B. DUARTE
Keyword(s):  
Shock Wave ◽  
Continuous Solution ◽  
Artificial Viscosity ◽  
Plane Shock ◽  
Hydrodynamic Equations ◽  
Von Neumann ◽  
One Dimensional ◽  
First Derivative ◽  
Spurious Oscillations ◽  
Steady Plane

Artificial viscosity is often expressed as a superposition of linear and quadratic terms in the first derivative of the velocity field. In trying to find a continuous solution for the hydrodynamic equations, we propose an alternative one-term artificial viscosity which is a linear form of the derivative of the specific volume. It is shown that this artificial viscosity is able to capture the profile of the steady plane shock wave, largely removing the non-physical oscillations originated by the artificial viscosity of von Neumann and Richtmyer. Analytical and numerical calculations for one-dimensional shock using both artificial viscosities are compared.

Effects of Shock Waves on Acceleration of Cell Growth Rate by Shock Tube

2008 ◽  
Author(s):  
Masaaki Tamagawa ◽  
Norikazu Ishimatsu
Keyword(s):  
Shock Wave ◽  
Growth Rate ◽  
Endothelial Cells ◽  
Shock Waves ◽  
Shock Tube ◽  
Test Case ◽  
Plane Shock ◽  
Extracorporeal Shock Wave ◽  
Shock Wave Pressure

This paper describes effects of shock waves on cells to certificate the angiogenesis by shock wave (pressure wave) in the clinical application such as ESW (Extracorporeal Shock Wave). Especially, to investigate the effects of shock waves on the endothelial cells in vitro, the cells worked by plane shock waves using shock tube apparatus are observed and measured in the microscope. The peak pressure working on the endothelial cells at the test case is 0.4 MPa. After working shock waves on suspended cells, growth rate (area per one cell and population of cells) are measured by image processing. It is found that the growth rate of the shock-worked cells from 0 to 4h is clearly high compared with control one. It is concluded that once shock waves worked, the cells have capacity to increase growth rate in vitro. This preliminary result will be applied to fundamental investigations about shock wave stimulus on several kinds of cells in future.

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